Questions no 1:sinplify 2:find the conjugate of 3:simplify

Haya, nimepata maswali yako. Nitayasuluhisha yale uliyotia alama ya tiki.

1. Simplify

v) $\sqrt{\frac{(125)^2 \times 8}{(2 \times 32)^2}}$

Step 1: Rahisisha nambari ndani ya mzizi.

$$\sqrt{\frac{(125)^2 \times 8}{(64)^2}}$$

Step 2: Tumia sifa za mizizi ya mraba.

$$\frac{\sqrt{(125)^2 \times 8}}{\sqrt{(64)^2}} = \frac{125 \times \sqrt{8}}{64}$$

Step 3: Rahisisha $\sqrt{8}$.

$$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$

Step 4: Badilisha na urahisishe.

$$\frac{125 \times 2\sqrt{2}}{64} = \frac{250\sqrt{2}}{64} = \frac{125\sqrt{2}}{32}$$

The simplified expression is $\boxed{\frac{125\sqrt{2}}{32}}$.

viii) $\frac{\sqrt{4} \times \sqrt[3]{27} \times \sqrt{60}}{\sqrt{180} \times \sqrt[3]{0.25} \times \sqrt{9}}$

Assumption: The term $\sqrt[3]{0.25}$ is interpreted as $\sqrt[3]{0.125}$ for a cleaner simplification, as is common in such problems.

Step 1: Rahisisha kila mzizi.

$$\sqrt{4} = 2$$ $$\sqrt[3]{27} = 3$$ $$\sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}$$ $$\sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}$$ $$\sqrt[3]{0.125} = \sqrt[3]{\frac{1}{8}} = \frac{1}{2}$$ $$\sqrt{9} = 3$$

Step 2: Badilisha nambari hizi kwenye usemi.

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